Numerische Mathematik

, Volume 121, Issue 4, pp 637–670 | Cite as

A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme



We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.

Mathematics Subject Classification (2000)

65M12 82D37 


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  1. 1.
    Alt H.W., Luckhaus S., Visintin A.: On nonstationary flow through porous media. Annali di Matematica Pura ed Applicata 136(1), 303–316 (1984)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arimburgo, F., Baiocchi, C., Marini, L.D.: Numerical approximation of the 1-D nonlinear drift-diffusion model in semiconductors. In: Nonlinear kinetic theory and mathematical aspects of hyperbolic system (Rapallo, 1992). Ser. Adv. Math. Appl. Sci., vol. 9, pp. 1–10. World Sci. Publ., River Edge, NJ (1992)Google Scholar
  3. 3.
    Brezzi F., Marini L.D., Pietra P.: Méthodes d’éléments finis mixtes et schéma de Scharfetter–Gummel. C. R. Acad. Sci. Paris Sér. I Math. 305(13), 599–604 (1987)MathSciNetMATHGoogle Scholar
  4. 4.
    Brezzi F., Marini L.D., Pietra P.: Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26(6), 1342–1355 (1989)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brézis H.: Analyse fonctionnelle: théorie et applications. Masson, Paris (1983)MATHGoogle Scholar
  6. 6.
    Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Carrillo J.A., Toscani G.: Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49(1), 113–142 (2000)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Chainais-Hillairet C., Filbet F.: Asymptotic behavior of a finite volume scheme for the transient drift-diffusion model. IMA J. Numer. Anal. 27(4), 689–716 (2007)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chainais-Hillairet C., Liu J.G., Peng Y.J.: Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis. M2AN 37(2), 319–338 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chainais-Hillairet C., Peng Y.J.: Convergence of a finite volume scheme for the drift-diffusion equations in 1-D. IMA J. Numer. Anal. 23, 81–108 (2003)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chainais-Hillairet C., Peng Y.J.: Finite volume approximation for degenerate drift-diffusion system in several space dimnesions. M3AS 14(3), 461–481 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    Courant R., Isaacson E., Rees M.: On the solution of nonlinear hyperbolic differential equations by finite differences. Commun. Pure. Appl. Math. 5, 243–255 (1952)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Eymard R., Fuhrmann J., Gärtner K.: A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102, 463–495 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Eymard R., Gallouët T.: H-convergence and numerical schemes for elliptic problems. SIAM J. Numer. Anal. 41(2), 539–562 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, vol. VII. Handb. Numer. Anal., vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000)Google Scholar
  16. 16.
    Eymard R., Gallouët T., Herbin R., Michel A.: Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92, 41–82 (2002)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Eymard R., Hilhorst D., Vohralík M.: A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105(1), 73–131 (2006)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Il’in A.M.: A difference scheme for a differential equation with a small parameter multiplying the highest derivative. Math. Zametki 6, 237–248 (1969)MATHGoogle Scholar
  19. 19.
    Jüngel A.: Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion. ZAMM 75(10), 783–799 (1995)MATHCrossRefGoogle Scholar
  20. 20.
    Jüngel A.: Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors. Math. Models Methods Appl. Sci. 5(4), 497–518 (1995)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Jüngel A., Pietra P.: A discretization scheme for a quasi-hydrodynamic semiconductor model. Math. Models Methods Appl. Sci. 7(7), 935–955 (1997)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Lazarov R.D., Mishev I.D., Vassilevski P.S.: Finite volume methods for convection–diffusion problems. SIAM J. Numer. Anal. 33(1), 31–55 (1996)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Markowich P.A., Ringhofer C.A., Schmeiser C.: Semiconductor equations. Springer-Verlag, Vienna (1990)MATHCrossRefGoogle Scholar
  24. 24.
    Markowich P.A.: The stationary semiconductor device equations, Springer edn. Computational Microelectronics, Vienna (1986)Google Scholar
  25. 25.
    Markowich P.A., Unterreiter A.: Vacuum solutions of the stationary drift-diffusion model. Ann. Scuola Norm. Sup. Pisa 20, 371–386 (1993)MathSciNetMATHGoogle Scholar
  26. 26.
    Scharfetter D.L., Gummel H.K.: Large signal analysis of a silicon Read diode. IEEE Trans. Elec. Dev. 16, 64–77 (1969)CrossRefGoogle Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques UMR 6620, CNRSUniversité Blaise PascalAubière CedexFrance

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